混合张量积、Capelli Berezinians 与 $\u200b\bggl(m|n)$ 的牛顿公式

The Capelli problem for $\mathfrak{gl}(m|n)$ and the spectrum of invariant differential operators

The "Capelli problem" for the symmetric pairs $(\mathfrak{gl}\times \mathfrak{gl},\mathfrak{gl})$ $(\mathfrak{gl},\mathfrak{o})$, and $(\mathfrak{gl},\mathfrak{sp})$ is closely related to the theory of Jack polynomials and shifted Jack polynomials for special values of the parameter. In this paper, we extend this connection to the Lie superalgebra setting, namely to the supersymmetric pairs $(\mathfrak{g},\mathfrak{g}):=(\mathfrak{gl}(m|2n),\mathfrak{osp}(m|2n))$ and $(\mathfrak{gl}(m|n)\times\mathfrak{gl}(m|n),\mathfrak{gl}(m|n))$, acting on $W:=S^2(\mathbb C^{m|2n})$ and $\mathbb C^{m|n}\oti…

通用Capelli恒等式和 queer 林格超代数的量子immanants

Universal Capelli identities and quantum immanants for the queer Lie superalgebra

本文通过构造 queer 林格超代数的量子immanants，证明了通用Capelli恒等式，并展示了其与阶乘Schur Q多项式的一致性。

The Capelli eigenvalue problem for Lie superalgebras

For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $\mathfrak g_\flat\cong \mathfrak g_\flat(-1)\oplus\mathfrak g_\flat(0)\oplus\mathfrak g_\flat(1)$, such that $\mathfrak g_\flat(-1)\cong J$. Set $V:=\mathfrak g_\flat(-1)^*$ and $\mathfrak g:=\mathfrak g_\flat(0)$. In most cases, the space $\mathcal P(V)$ of superpolynomials on $V$ is a completely reducible and multiplicity-free representation of $\mathfrak g$, with a decomposition $\mathcal P(V):=\bigoplus_{λ\inΩ}V_λ$, where $\left(V_λ\right)_{λ\inΩ}…

关于李超代数$\boldsymbol{\frak{gl}_{m|n}}$的双Yangian的量子Berezinian的注记

A note on the quantum Berezinian for the double Yangian of the Lie superalgebra $\mathfrak{gl}_{m|n}$

该研究将量子Berezinian概念推广到李超代数$\frak{gl}_{m|n}$的双Yangian ${\rm DY}(\frak{gl}_{m|n})$，证明其系数是该双Yangian中心的代数独立拓扑生成元。

Schur $Q$-functions and the Capelli eigenvalue problem for the Lie superalgebra $\mathfrak q(n)$

Let $\mathfrak l:= \mathfrak q(n)\times\mathfrak q(n)$, where $\mathfrak q(n)$ denotes the queer Lie superalgebra. The associative superalgebra $V$ of type $Q(n)$ has a left and right action of $\mathfrak q(n)$, and hence is equipped with a canonical $\mathfrak l$-module structure. We consider a distinguished basis $\{D_λ\}$ of the algebra of $\mathfrak l$-invariant super-polynomial differential operators on $V$, which is indexed by strict partitions of length at most $n$. We show that the spectrum of the operator $D_λ$, when it acts on the algebra $\mathscr P(V)$ of super-polynomials on $V$, …

Mixed tensor products and Capelli-type determinants

In this paper we study properties of a homomorphism $ρ$ from the universal enveloping algebra $U=U(\mathfrak{gl}(n+1))$ to a tensor product of an algebra $\mathcal D'(n)$ of differential operators and $U(\mathfrak{gl}(n))$. We find a formula for the image of the Capelli determinant of $\mathfrak{gl}(n+1)$ under $ρ$, and, in particular, of the images under $ρ$ of the Gelfand generators of the center $Z(\mathfrak{gl}(n+1))$ of $U$. This formula is proven by relating $ρ$ to the corresponding Harish-Chandra isomorphisms, and, alternatively, by using a purely computational approach. Furthermore, we…